MOSCOW, July 18 (Reuters) - A group of Soviet-born tycoons said on Wednesday it had notified BP of its intention to start talks with the British oil major on increasing its stake in Russia’s third-largest crude producer, TNK-BP.

BP put its stake in TNK-BP up for sale on June 1 after a breakdown in relations with the AAR consortium, with which it shares control over a firm that accounts for almost a third of BP’s total crude output.

“AAR and BP both realise that a fundamental realignment in the ownership of TNK-BP is necessary in order to allow each of the shareholders to achieve its strategic objectives and eliminate the internal contradictions that are preventing further development of TNK-BP,” AAR CEO Stan Polovets said in a statement.

“We hope that BP will now engage with us constructively, particularly given the importance of TNK-BP to BP’s overall position.”

Under a timeline set by TNK-BP’s shareholder agreement, AAR had until the end of this week to express its interest in buying BP’s stake. BP can negotiate with other interested parties but cannot do a deal with them for a further 90 days.

In London, BP confirmed it would enter “good faith” negotiations with AAR on the sale of part or all of its stake in TNK-BP.

“BP is also able to and will enter into negotiations with other interested parties in parallel for the sale of its share in TNK-BP,” BP said in a news release.

“There can be no guarantee that any transaction will take place,” BP also said.

AAR said that BP had not asked for its approval to provide confidential information to any other prospective buyers in the seven weeks since the British firm announced its intention to sell seven weeks ago.

AAR also confirmed it had made a proposal in May, prior to BP’s public announcement, to buy one half of BP’s stake in TNK-BP - or 25 percent - at what it called a current market price.

Polovets told Reuters last Sunday that AAR would be willing to acquire 25 percent of TNK-BP at a current market price of around $10 billion.

But the quartet of billionaires, led by retail-to-banking entrepreneur Mikhail Fridman, would rather resolve their conflict with BP by selling out for cash and a stake in the British oil company, Polovets also said.

Tensions between the tycoons and BP flared after the London-based company attempted to partner with Russia’s top crude producer Rosneft last year. AAR won a court injunction against the deal, which BP later abandoned.

Currently, an arbitration panel operating under British law is studying AAR’s request to rule on whether BP breached the TNK-BP shareholders’ agreement by attempting to partner with Rosneft.

An arbitration ruling in favour of AAR could make BP liable to billions of dollars in damages, AAR has said.

“A sale by BP of all or any part of its interest in TNK-BP will not affect BP’s potential liability to AAR and TNK-BP for breach of the shareholders’ agreement in connection with BP’s attempted strategic alliance with Rosneft,” AAR said.